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Introduction to STATICS and DYNAMICS F 1 F 2 N 1 N 2 F s M s ˆ ı ˆ ˆ k Andy Ruina and Rudra Pratap Pre-print for Oxford University Press, January 2002 Summary of Mechanics 0) The laws of mechanics apply to any collection of material or ‘body.’ This body could be the overall system of study or any part of it. In the equations below, the forces and moments are those that show on a free body diagram. Interacting bodies cause equal and opposite forces and moments on each other. I) Linear Momentum Balance (LMB)/Force Balance Equation of Motion F i = ˙ L The total force on a body is equal to its rate of change of linear momentum. (I) Impulse-momentum (integrating in time) t 2 t 1 F i ·dt = L Net impulse is equal to the change in momentum. (Ia) Conservation of momentum (if F i = 0 ) ˙ L = 0 ⇒ L = L 2 − L 1 = 0 When there is no net force the linear momentum does not change. (Ib) Statics (if ˙ L is negligible) F i = 0 If the inertial terms are zero the net force on system is zero. (Ic) II) Angular Momentum Balance (AMB)/Moment Balance Equation of motion M C = ˙ ˙ H C The sum of moments is equal to the rate of change of angular momentum. (II) Impulse-momentum (angular) (integrating in time) t 2 t 1 M C dt = H C The net angular impulse is equal to the change in angular mo mentum. (IIa) Conservation of angular momentum (if M C = 0) ˙ H C = 0 ⇒ H C = H C 2 − H C 1 = 0 If there is no net moment about point C then the angular momentum about point C does not change. (IIb) Statics (if ˙ H C is negligible) M C = 0 If the inertial terms are zero then the total moment on the system is zero. (IIc) III) Power Balance (1st law of thermodynamics) Equation of motion ˙ Q + P = ˙ E K + ˙ E P + ˙ E int ˙ E Heat flow plus mechanical power into a system is equal to its change in energy (kinetic + potential + internal). (III) for finite time t 2 t 1 ˙ Qdt + t 2 t 1 Pdt = E The net energy flow goingin is equal to the net change in energy. (IIIa) Conservation of Energy (if ˙ Q = P = 0) ˙ E = 0 ⇒ E = E 2 − E 1 = 0 If no energy flows into a system, then its energy does not change. (IIIb) Statics (if ˙ E K is negligible) ˙ Q + P = ˙ E P + ˙ E int If there is no change of kinetic energy then the change of potential and internal energy is due to mechanical work and heat flow. (IIIc) Pure Mechanics (if heat flow and dissipation are negligible) P = ˙ E K + ˙ E P In a system well modeled as purely mechanical the change of kinetic and potential energy is due to mechanical work. (IIId) Some Definitions r or x Position .e.g., r i ≡ r i/O is theposition of apoint i relative to the origin, O) v ≡ d r dt Velocity .e.g., v i ≡ v i/O is the velocity ofa point i relativeto O, measured in anon-rotating reference frame) a ≡ d v dt = d 2 r dt 2 Acceleration .e.g., a i ≡ a i/O is the acceleration of a point i relative to O, measured in a New- tonian frame) ω Angular (Please also look at the tables inside the back cover.) velocity A measure ofrotational velocityof arigid body. α ≡ ˙ ω Angular acceleration A measure of rotational acceleration of a rigid body. L ≡ m i v i discrete vdm continuous Linear momentum A measure of a system’s net translational rate (weighted by mass). = m tot v cm ˙ L ≡ m i a i discrete adm continuous Rate of change of linear momentum The aspect of motion thatbalances thenet force on a system. = m tot a cm H C ≡ r i/C × m i v i discrete r /C × vdm continuous Angular momentum about point C A measure of the rotational rate of a sys- tem about a point C (weighted by mass and distance from C). ˙ H C ≡ r i/C × m i a i discrete r /C × adm continuous Rate of change of angular mo- mentum about point C The aspect of motion thatbalances thenet torque on a system about a point C. E K ≡ 1 2 m i v 2 i discrete 1 2 v 2 dm continuous Kinetic energy A scalar measure of net system motion. E int = (heat-like terms) Internal energy The non-kinetic non-potential part of a system’s total energy. P ≡ F i · v i + M i · ω i Power of forces and torques The mechanical energy flow into a sys- tem. Also, P ≡ ˙ W, rate of work. [I cm ]≡ I cm xx I cm xy I cm xz I cm xy I cm yy I cm yz I cm xz I cm yz I cm zz Moment ofinertia matrixabout cm A measure of how mass is distributed in a rigid body. c Rudra Pratap and Andy Ruina, 1994-2002. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, or otherwise, without prior written permission of the authors. This book is a pre-release version of a book in progress for Oxford University Press. Acknowledgements. The following are amongst those who have helped with this book as editors, artists, tex programmers, advisors, critics or suggestors and cre- ators of content: Alexa Barnes, Joseph Burns, Jason Cortell, Ivan Dobrianov, Gabor Domokos,MaxDonelan, ThuDong, GailFish,MikeFox,JohnGibson, RobertGhrist, Saptarsi Haldar, Dave Heimstra, Theresa Howley, Herbert Hui, Michael Marder, Elaina McCartney, Horst Nowacki, Arthur Ogawa, Kalpana Pratap, Richard Rand, Dane Quinn, Phoebus Rosakis, Les Schaeffer, Ishan Sharma, David Shipman, Jill Startzell, Saskya van Nouhuys, Bill Zobrist. Mike Coleman worked extensively on the text, wrote many of the examples and homework problems and created many of the figures. David Ho has drawn or improved most of the computer art work. Some of the homework problems are modifications from the Cornell’s Theoretical and Ap- plied Mechanics archives and thus are due to T&AM faculty or their libraries in ways that we do not know how to give proper attribution. Our editor Peter Gordon has been patient and supportive for too many years. Many unlisted friends, colleagues, relatives, students, and anonymous reviewers have also made helpful suggestions. Software used to prepare this book includes TeXtures, BLUESKY’s implementation of LaTeX, Adobe Illustrator, Adobe Streamline, and MATLAB. Most recent text modifications on January 29, 2002. Introduction to STATICS and DYNAMICS F 1 F 2 N 1 N 2 F s M s ˆ ı ˆ ˆ k Andy Ruina and Rudra Pratap Pre-print for Oxford University Press, January 2002 Contents Preface iii To the student viii 1 What is mechanics? 1 2 Vectors for mechanics 7 2.1 Vector notation and vector addition 8 2.2 The dot product of two vectors 23 2.3 Cross product, moment, and moment about an axis 32 2.4 Solving vector equations 50 2.5 Equivalent force systems 69 3 Free body diagrams 79 3.1 Free body diagrams 80 4 Statics 107 4.1 Static equilibrium of one body 109 4.2 Elementary truss analysis 129 4.3 Advanced truss analysis: determinacy, rigidity, and redundancy 138 4.4 Internal forces 146 4.5 Springs 162 4.6 Structures and machines 179 4.7 Hydrostatics 195 4.8 Advanced statics 207 5 Dynamics of particles 217 5.1 Force and motion in 1D 219 5.2 Energy methods in 1D 233 5.3 The harmonic oscillator 240 5.4 More on vibrations: damping 257 5.5 Forced oscillations and resonance 264 5.6 Coupled motions in 1D 274 5.7 Time derivative of a vector: position, velocity and acceleration 281 5.8 Spatial dynamics of a particle 289 5.9 Central-force motion and celestial mechanics 304 5.10 Coupled motions of particles in space 314 6 Constrained straight line motion 329 6.1 1-D constrained motion and pulleys 330 6.2 2-D and 3-D forces even though the motion is straight 343 i ii CONTENTS 7 Circular motion 359 7.1 Kinematics of a particle in planar circular motion 360 7.2 Dynamics of a particle in circular motion 371 7.3 Kinematics of a rigid body in planar circular motion 378 7.4 Dynamics of a rigid body in planar circular motion 395 7.5 Polar moment of inertia: I cm zz and I O zz 410 7.6 Using I cm zz and I O zz in 2-D circular motion dynamics 420 8 General planar motion of a single rigid body 437 8.1 Kinematics of planar rigid-body motion 438 8.2 General planar mechanics of a rigid-body 452 8.3 Kinematics of rolling and sliding 467 8.4 Mechanics of contacting bodies: rolling and sliding 480 8.5 Collisions 500 9 Kinematics using time-varying base vectors 517 9.1 Polar coordinates and path coordinates 518 9.2 Rotating reference frames and their time-varying base vectors 532 9.3 General expressions for velocity and acceleration 545 9.4 Kinematics of 2-D mechanisms 558 9.5 Advance kinematics of planar motion 572 10 Mechanics of constrained particles and rigid bodies 581 10.1 Mechanics of a constrained particle and of a particle relative to a moving frame 584 10.2 Mechanics of one-degree-of-freedom 2-D mechanisms 602 10.3 Dynamics of rigid bodies in multi-degree-of-freedom 2-D mechanisms618 11 Introduction to three dimensional rigid body mechanics 637 11.1 3-D description of circular motion 638 11.2 Dynamics of fixed-axis rotation 648 11.3 Moment of inertia matrices 661 11.4 Mechanics using the moment of inertia matrix 672 11.5 Dynamic balance 693 A Units and dimensions 701 A.1 Units and dimensions 701 B Contact: friction and collisions 711 B.1 Contact laws are all rough approximations 712 B.2 Friction 713 B.3 A short critique of Coulomb friction 716 B.4 Collision mechanics 721 Homework problems 722 Answers to *’d problems 831 Index 837 Preface This is a statics and dynamics text for second or third year engineering students with an emphasis on vectors, free body diagrams, the basic momentum balance principles, and the utility of computation. Students often start a course like this thinking of mechanics reasoning as being vague and complicated. Our aim is to replace this loose thinking with concrete and simple mechanics problem-solving skills that live harmoniously with a useful mechanical intuition. Knowledge of freshman calculus is assumed. Although most students have seen vector dot and cross products, vector topics are introduced from scratch in the context of mechanics. The use of matrices (to tidy-up systems of linear equations) and of differential equations (for describing motion in dynamics) are presented to the extent needed. The set-up of equations for computer solutions is presented in a pseudo- language easily translated by a student into one or another computation package that the student knows. Organization We have aimed here to better unify the subject, in part, by an improved organization. Mechanics can be subdivided in various ways: statics vs dynamics, particles vs rigid bodies, and 1 vs 2 vs 3 spatial dimensions. Thus a 12 chapter mechanics table of contents could look like this I. Statics A. particles 1) 1D 2) 2D 3) 3D B. rigid bodies 4) 1D 5) 2D 6) 3D II. Dynamics C. particles 7) 1D 8) 2D 9) 3D D. rigid bodies 10) 1D 11) 2D 12) 3D complexity of objects number of dimensions how much inertia 1D 2D 3D static dynamic particle rigid body However,thesetopicsare farfrom equalintheirdifficultyorinthenumberof subtopics they contain. Further, there are various concepts and skills that are common to many of the 12 sub-topics. Dividingmechanics into these bits distracts from the unity of the subject. Although some vestiges of the scheme above remain, our book has evolved to a different organization through trial and error, thought and rethought, review and revision, and nine semesters of student testing. The first four chapters cover the basics of statics. Dynamics of particles and rigid bodies, based on progressively more difficult motions, is presented in chapters five to eleven. Relatively harder topics, that might be skipped in quicker courses, are identifiable by chapter, section or subsection titles containing words like “three dimensional” or “advanced”. In more detail: iii iv PREFACE Chapter 1 defines mechanics as a subject which makes predictions about forces and motions using models ofmechanical behavior, geometry, and the basic balance laws. The laws of mechanics are informally summarized. Chapter 2 introduces vector skills in the context of mechanics. Notational clarity is emphasized because correct calculation is impossible without distinguishing vectors from scalars. Vector addition is motivated by the need to add forces and relative positions, dot products are motivated as the tool which reduces vector equations to scalar equations, and cross products are motivated as the formula which correctly calculates the heuristically motivated concept of moment and moment about an axis. Chapter 3 is about freebody diagrams. Itis aseparatechapterbecause, inour experience, good use of free body diagrams is almost synonymous with correct mechanics problem solution. To emphasize this to students we recommend that, to get any credit for a problem that uses balance laws in the rest of the course, a good free body diagram must be drawn. Chapter 4 makes up a short course in statics including an introduction to trusses, mecha- nisms, beams and hydrostatics. The emphasis is on two-dimensional problems until the last, more advanced section. Solution methods that depend on kine- matics (i.e., work methods) are deferred until the dynamics chapters. But for the stretch of linear springs, deformations are not covered. Chapter 5 is about unconstrained motion of one or more particles. It shows how far you can go using F = m a and Cartesian coordinates in 1, 2 and 3 dimensions in the absence of kinematic constraints. The first five sections are a thor- ough introduction to motion of one particle in one dimension, so called scalar physics, namely the equation F(x,v,t) = ma and special cases thereof. The chapter includes some review of freshman calculus as well as an introduction to energy methods. A few special cases are emphasized, namely, constant ac- celeration, force dependent on position (thus motivating energy methods), and the harmonic oscillator. After one section on coupled motions in 1 dimension, sections seven to ten discuss motion in two and three dimensions. The easy set up for computation of trajectories, with various force laws, and even with multiple particles, is emphasized. The chapter ends with a mostly theoretical section on the center-of-mass simplifications for systems of particles. Chapter 6 is the first chapter that concerns kinematic constraint in its simplest context, systems that are constrained to move without rotation in a straight line.In one dimension pulley problems provide the main example. Two and three dimensional problems are covered, such as finding structural support forces in accelerating vehicles and the slowing or incipient capsize of a braking car. Angular momentum balance is introduced as a needed tool but without the usual complexities of curvilinear motion. Chapter 7 treats pure rotation about a fixed axis in two dimensions. Polar coordinates and base vectors are first used here in their simplest possible context. The primary applications are pendulums, gear trains, and rotationally accelerating motors or brakes. Chapter 8 treats general planarmotion of a (planar) rigidbody includingrolling, sliding and free flight. Multi-body systems are also considered so long as they do not involve constraint (i.e., collisions and spring connections but not hinges or prismatic joints). Chapter 9 is entirely about kinematics of particle motion. The over-riding theme is the use of base vectors which change with time. First, the discussionof polar coor- dinates started in chapter7 is completed. Then pathcoordinates are introduced. The kinematics of relative motion, a topic that many students find difficult, is treated carefully but not elaborately in two stages. First using rotating base PREFACE v vectors connected to a moving rigid body and then using the more abstract notation associated with the famous “five term acceleration formula.” Chapter 10 is about the mechanics of particles and rigid bodies utilizing the relative mo- tion kinematics ideas from chapter 9. This is the capstone chapter for a two- dimensional dynamics course. After this chapter a good student should be able to navigate through and use most of the skills in the concept map on page 582. Chapter 11 is an introduction to 3D rigid body motion. It extends chapter 7 to fixed axis rotation in three dimensions. The key new kinematic tool here is the non- trivial use of the cross product for calculating velocities and accelerations. Fixed axis rotation is the simplest motion with which one can introduce the full moment of inertia matrix, where the diagonal terms are analogous to the scalar 2D moment of inertia and the off-diagonal terms have a “centripetal” interpretation. Themainnewapplication isdynamicbalance. Inour experience going past this is too much for most engineering students in the first mechanics course after freshman physics, so the book ends here. Appendix A on units and dimensions is for reference. Because students are immune to preaching about units out of context, such as in an early or late chapter like this one, the mainmessages arepresented by example throughout the book (the book may be unique amongst mechanics texts in this regard): – All engineering calculations using dimensionalquantities must be dimen- sionally ‘balanced’. – Units are ‘carried’ from one line of calculation to the next by the same rules as go numbers and variables. Appendix B oncontact laws(frictionand collisions)isfor referencefor studentswhopuzzle over these issues. A leisurely onesemester statics course, or amore fast-paced half semesterprelude to strength of materials should use chapters 1-4. A typical one semester dynamics course should cover most of of chapters 5-11 preceded by topics from chapters 1-4, as needed. A one semesterstatics and dynamics course should cover about two thirds of chapters 1-6 and 8. A full year statics and dynamics course should cover most of the book. Organization and formatting Each subject is covered in various ways. • Every section starts with descriptive text and short examples motivating and describing the theory; • More detailed explanations of the theory are in boxes interspersed in the text. For example, one box explains the common derivation of angular momentum balance form linear momentum balance, one explains the genius of the wheel, and another connects ω based kinematics to ˆ e r and ˆ e θ based kinematics; • Sample problems (marked witha gray border)at theend of mostsections show how to do homework-like calculations. These set an example to the student in their consistent use of free body diagrams, systematic application of basic principles, vectornotation, units, andchecks against intuition and specialcases; • Homework problems at the end of each chapter give students a chance to practice mechanics calculations. The first problems for each section build a student’s confidence with the basic ideas. The problems are ranked in approxi- mate order of difficulty, with theoretical questions last. Problems marked with an * have an answer at the back of the book; [...]... ability to routinely evaluate standard functions (x 3 , cos−1 θ, etc.), to enter and manipulate lists and arrays of numbers, and to write short programs Classical languages, applied packages, and simulators Programming in standard languages such as Fortran, Basic, C, Pascal, or Java probably take too much time to use in solving simple mechanics problems Thus an engineer needs to learn to use one or... mechanics and, in particular, the concept of vector addition always has a physical interpretation 2.1 Vector notation and vector addition Facility with vectors has several aspects 1 You must recognize which quantities are vectors (such as force) and which are scalars (such as length) 2 You have to use a notation that distinguishes between vectors and scalars using, for example, a, or a for acceleration... the arrow is only schematic, the magnitude and direction are determined by the algebraic symbol F It is sometimes helpful to draw the arrow in the direction of the vector and approximately to scale, but this is not necessary 2.1 Vector notation and vector addition 13 Graphical: a scalar multiplies an arrow Indicate a vector’s direction by drawing an arrow with direction indicated by marked angles or... system causes it to rotationally accelerate (angular momentum balance), and III) The change of energy of a system is due to the energy flow into the system (energy balance) The principles of action and reaction, linear momentum balance, angular momentum balance, and energy balance, are actually redundant various ways Linear momentum balance can be derived from angular momentum balance and, sometimes (see... vector F But bold face is inconvenient for hand written work The lack of bold face pens and pencils tempts students to transcribe a bold F as F But F with no adornment represents a scalar and not a vector Learning how to work with vectors and scalars is hard enough without the added confusion of not being able to tell at a glance which terms in your equations are vectors and which are scalars F Underlining... with names like ‘Lagrange’s equations,’ ‘Hamilton’s equations,’ ‘virtual work’, and ‘variational principles.’ Should you take an interest in theoretical mechanics, you may learn these approaches in more advanced courses and books, most likely in graduate school 5 Statics, dynamics, and strength of materials Elementary mechanics is traditionally partitioned into three courses named statics , dynamics ,... readers What’s in your toolbox? In the toolbox of someone who can solve lots of mechanics problems are two well worn tools: • A vector calculator that always keeps vectors and scalars distinct, and • A reliable and clear free body diagram drawing tool Because many of the terms in mechanics equations are vectors, the ability to do vector calculations is essential Because the concept of an isolated system... similar words kinematics and kinetics Kinematics concerns geometry with no mention of force and kinetics concerns the relation of force to motion Once one has mastered statics, the hard part of dynamics is the kinematics Dynamics is useful for the analysis of, for example, fast machines, vibrations, and ballistics Strength of materials expands statics to include material properties and also pays more attention... a skateboard and rider, a quartz timing crystal, a building in an earthquake, a piano string, and a space shuttle Will a wheel slip? a gear tooth break? a car tip over? What muscles are used when you hit a key on your computer? How do people balance on skateboards? Which buildings are more likely to fall in what kinds of earthquakes? Why are low pitch piano strings made with helical windings instead... (Filename:sfig1.2. 4a) SAMPLE 2.2 Drawing a vector from its length and direction: A vector r is 3.6 ft long and is directed 33.7o from the x-axis towards the negative y-axis Draw r Solution We first draw the x and y axes and then draw r as a line from the origin at an angle −33.7o from the x-axis (minus sign means measuring clockwise), measure 3.6 units (magnitude of r ) along this line and finally put an arrowhead pointing . The primary applications are pendulums, gear trains, and rotationally accelerating motors or brakes. Chapter 8 treats general planarmotion of a (planar) rigidbody. Heimstra, Theresa Howley, Herbert Hui, Michael Marder, Elaina McCartney, Horst Nowacki, Arthur Ogawa, Kalpana Pratap, Richard Rand, Dane Quinn, Phoebus Rosakis,